Find the inverse function of y = (3 ^ x-3 ^ - x) / 2

Find the inverse function of y = (3 ^ x-3 ^ - x) / 2

Let 3 ^ x = t, then: T > 0, then: y = (t-1 / T) / 2t-1 / T = 2yt & # 178; - 2yt-1 = 0, the root formula is: T = [2Y ± 2 √ (Y & # 178; + 1)] / 2 = y ± √ (Y & # 178; + 1) because t > 0, so, t = y + √ (Y & # 178; + 1), that is; 3 ^ x = y + √ (Y & # 178; + 1) so, x = log3 [y + √ (Y & # 178; + 1)]

Inverse function of y = (x + (1 + x ^ 2) ^ 1 / 2) ^ 1 / 3 + (x - (1 + x ^ 2) ^ 1 / 2) ^ 1 / 3

y^3=2x+3y{[x+√(1+x^2)][x-√(1+x^2)]}^(1/3)
=2x-3y,
∴x=(y^3+3y)/2,
x. Y = (x ^ 3 + 3x) / 2

If a store buys a kind of clothes with a unit price of 40 yuan and sells them at 50 yuan per piece, it can sell 500 pieces per month. If each piece is increased by 1 yuan, the monthly sales will be reduced by 10 pieces
The increase of unit price X Yuan is a function of profit y
Is 28000 yuan the maximum profit? If not, please answer when is the maximum profit

（1）
y=(50+x-40)(500-10x)
=(10+x)(500-10x)
=-10x²+400x+5000
（2）
no
The axis of symmetry is x = 400 / 20 = 20
When x = 20
Profit y has a maximum
=-10*400+8000+5000
=9000
Maximum profit = 9000 yuan

A mathematical quadratic function problem
The quadratic function y = x & # 178; - 4x + 3 is known. The symmetric axis and vertex coordinates of the parabola y = x & # 178; - 4x + 3 are obtained by the collocation method. Let the parabola intersect the X axis at two points a and B, and the parabola intersect the Y axis at point C, and the area of △ ABC be calculated

∵ y = x & # 178; - 4x + 3 = x & # 178; - 4x + 4-4 + 3 = (X-2) &# 178; - 1 ∵ the vertex coordinate of the parabola is (2, - 1) let x = 0, then y = 3 ∵ C (0,3) let y = 0, then x & # 178; - 4x + 3 = 0 (x-1) (x-3) = 0x-1 = 0, or x-3 = 0 ∵ X1 = 1, X2 = 3 ∵ the intersection of the parabola and the X axis is a (1,0), B (3,0), then AB = 2 ? s ?

Find the range of function f (x) = x & # 178; - 2ax-1 in X ∈ [0,2];
Why is it divided into
;❷0≤a≤1
;❸1＜a≤2
How about the discussion of the four cases, especially the two cases, I have never understood
I didn't expect that kind of character is useless. I want to ask the middle two items. I don't understand why I want to discuss them in that way

Find the value range of function f (x) = x & # 178; - 2ax-1 in X ∈ [0,2]; why is it divided into four cases: F (x) = x & # 178; - 2ax-1 in the range of X ∈ [0,2]; why is it divided into four cases: F (x) = x & # 10102; a < 0; - # 10103; 0 ≤ a ≤ 1; - # 10104; 1 < a ≤ 2; - # 10105; a > 2

Symmetry axis and vertex of square opening direction of y = 1 - (half-x) and solution method~

y=x²-x+3/4
y=(x-1/2)²+1/2
Opening upward, axis of symmetry x = 1 / 2, vertex (1 / 2,1 / 2)

Draw the image of function y = x2 + 2x, y = x2 + 2 | x | in the same coordinate system

For the first function, we can learn from the knowledge of junior high school, with the help of vertex coordinates, opening direction, and coordinate axis intersection coordinates; formula y = x2 + 2x = (x + 1) 2-1, so the symmetry axis of the function is x = - 1, fixed point is (- 1, - 1), parabola opening upward, as shown in the figure

We know the linear function y = 2X-4 and y = - x + 2. (1) draw their images in the agreed coordinate system. (2) find their images
(2) Find out the area of their image and y-axis

y=2x-4
The line between point (2,0) and point (0, - 4) is the image of this function
y=-x+2
The line between point (2,0) and point (0,2) is the image of this function
The two lines and the y-axis form a triangle,
The base of triangle is the distance between point (0,2) and point (0, - 4) = 6
The height of the triangle is the distance from the point (2,0) to the origin = 2
The area of triangle is 1 / 2 × 6 × 2 = 6

In the given rectangular coordinate system, draw the image of the first-order function y = 1-2x, translate the line y = 1-2x upward by 2 units, and find the analytical formula of the line after translation

y=1-2x
=-2x+1,k=-2,b=1
y =-2x+1+2
=-2x+3
Up translation 2 units, that is B + 2, so y = - 2x + 3

The function y = - 2x + 4 is drawn in the coordinate system by using the point tracing method
(to list)

x y |
（0,4） |
（2,0） |